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Effect of Time Delays in CharacterizingContinuous Mixing of Aqueous Xanthan GumSolutionsVishal Kumar PatelRyerson University

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Recommended CitationPatel, Vishal Kumar, "Effect of Time Delays in Characterizing Continuous Mixing of Aqueous Xanthan Gum Solutions" (2010). Thesesand dissertations. Paper 1477.

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EFFECT OF TIME DELAYS IN CHARACTERIZING CONTINUOUS MIXING

OF AQUEOUS XANTHAN GUM SOLUTIONS

By

Vishal Kumar R. Patel

Bachelor of Engineering in Chemical Engineering

Gujarat University, June 2000

A THESIS

PRESENTED TO RYERSON UNIVERSITY

IN PARTIAL FULFILMENT OF THE

REQUIREMENTS FOR THE DEGREE OF

MASTER OF APPLIED SCIENCE

IN THE PROGRAM OF

CHEMICAL ENGINEERING

TORONTO, ONTARIO, CANADA, 2010

© Vishal Kumar R. Patel, 2010

iii

AUTHOR’S DECLARATION

I hereby declare that I am the sole author of this Thesis or major Research Paper.

I authorize Ryerson University to lend this thesis or Major Research Paper to other

institutions or individuals for the purpose of scholarly research.

------------------------------------

I further authorize Ryerson University to reproduce this thesis or dissertation by

photocopying or by other means, in total or in part, at the request of other institutions or

individuals for the purpose of scholarly research.

------------------------------------

v

EFFECT OF TIME DELAYS IN CHARACTERIZING CONTINUOUS MIXING

OF AQUEOUS XANTHAN GUM SOLUTIONS

Master of Applied Science

In the Program of

Chemical Engineering

2010


School of Graduate Studies

Ryerson University

ABSTRACT

Aqueous xanthan gum solutions are non-Newtonian fluids, pseudoplastic fluids

possessing yield stress. Their continuous mixing is an extremely complicated

phenomenon exhibiting non idealities such as channeling, recirculation and stagnation.

To characterize the continuous mixing of xanthan gum solutions, three dynamic models

were utilized: (1) a dynamic model with 2 time delays in discrete time domain, (2) a

dynamic model with 2 time delays in continuous time domain, and (3) a simplified

dynamic model with 1 time delay in discrete time domain. A hybrid genetic algorithm

was employed to estimate the model parameters through the experimental input-output

dynamic data. The extents of channeling and fully-mixed volume were used to compare

the performances of these three models. The dynamic model parameters exerting strong

influence on the model response were identified. It was observed that the models with 2

time delays gave a better match with the experimental results.

Keywords: Continuous mixing, Dynamic model, Non-Newtonian fluid, Parameter

identification, Hybrid genetic algorithm.

vii

ACKNOWLEDGEMENTS

I heartily thank my supervisors, Dr. Farhad Ein-Mozaffari and Dr. Simant R. Upreti,

whose encouragement, supervision and support from the preliminary to the concluding

level enabled me to develop an understanding of the subject.

Also it is a pleasure to thank those who made this thesis possible such as my father

Ramsingbhai G. Patel and my uncle Ratanjibhai B. Patel, who gave me moral support

and inspire me to complete the challenging research that lies behind it.

The financial support provided by Natural Science and Engineering Research Council of

Canada (NSERC) for funding this research is greatly acknowledged.


ix

CONTENTS

AUTHOR’S DECLARATION………………………………………………….. iii

ABSTRACT.....…………………………….…………………………………….. v

ACKNOWLEDGEMENTS...……………………………………………………. vii

LIST OF FIGURES.……………………………………………………………… xii

LIST OF TABLES…..…………………………………………………………… xiii

NOMENCLATURE……………………………………………………………... xiv

1 INTRODUCTION……….………………………………...…………………… 1

1.1 Background………………………………………………………………… 1

1.2 Thesis objective and contributions………………………………………… 2

1.2.1 Problem statement………………………..…….…………………. 2

1.2.2 Dynamic models..……………………...…………………………. 2

1.2.3 Hybrid genetic algorithm……………..…….…………………….. 2

1.3 Thesis layout……………………………………………….……………….. 3

2 MIXING………………..…………………………………………………….…. 5

2.1 Introduction……………..………………………………….……………... 5

2.1.1 Batch mixing………………………..……………….…………….. 5

2.1.2 Continuous mixing…………………………..……….……………. 6

2.2 Continuous mixing of non-Newtonian fluids…………….….………….… 8

2.2.1 Introduction……………………….………………….……….…… 8

2.2.2 Rheology of non-Newtonian fluid………..………………….……. 8

2.2.3 Literature review on continuous mixing of

pseudoplastic with yield stress fluids..…………………………… 12

3 MATHEMATICAL MODEL...…………………..…………………………… 15

3.1 Introduction……………………………………………………………….. 15

3.2 Dynamic models for continuous mixing of non-Newtonian fluids.………. 16

3.2.1 Ein-Mozaffari‟s dynamic model…………………………………… 16

3.2.2 Patel‟s dynamic model…………………………………………….. 18

3.2.3 Soltanzadeh‟s dynamic model…………………………………….. 19

x

4 SYSTEM IDENTIFICATION…………..…………………….………………... 21

4.1 Introduction…………………………………………………………………. 21

4.2 Estimation of parameters via hybrid genetic algorithm…………………....... 23

4.2.1 Discrete time domain approach……………………………………… 25

4.2.2 Continuous time domain approach…………………………………... 29

4.3 Sensitivity analysis…………………………………………………………. 31

5 GENETIC ALGORITHM……………………………………………………… 33

5.1 Genetic algorithm……………………………………………………………. 33

5.1.1 Introduction……………………………………………………………33

5.2 Hybrid genetic algorithm…………………………………………………… 37

5.2.1 Introduction…………………………………………………………. 37

5.2.2 Steps of algorithm……………………………………………………. 42

5.3 Experimental overview……………………………………………………… 45

5.4 Rheology of xanthan gum.……………………………………………............ 47

6 RESULTS AND DISCUSSION…………………………………………………. 49

6.1 Model validation……………………………………………………………. 49

6.2 Model predicted output..……………………………………………………. 53

6.3 Effect of impeller speeds on extent of channeling and fully mixed volume.. 55

6.4 Effect of mass concentrations on extent of channeling and

Fully mixed volume……………………………………………………….. 59

6.5 Effect of fluid flow rates on extent of channeling and fully mixed volume . 64

6.6 Effect of output locations on extent of channeling and fully mixed volume 66

6.7 Sensitivity analysis…………………………………………………………. 69

7 CONCLUSION AND RECOMMENDATION FOR FUTURE WORK …..…73

7.1 Conclusion…………………………………………………………………. 73

7.2 Recommendation for future work.…………………………………………. 74

xi

REFERENCES……………………………………………………………………. 75

APPENDIX………………………………………………………………………… 79

xii

LIST OF FIGURES:

Figure 1 Categories of non-Newtonian fluids………………………………………….. 9

Figure 2 Block diagram of Ein-Mozaffari‟s dynamic model…………………………... 16

Figure 3 Block diagram of Patel‟s dynamic model……………………………………. 17

Figure 4 Block diagram of Soltanzadeh‟s dynamic model……………………………. 19

Figure 5 Flow diagram of system identification……………………………………….. 20

Figure 6 Standard genetic algorithm…………………………………………………… 33

Figure 7 Hybrid genetic algorithm……………………………………………………... 38

Figure 8 Schematic of experimental setup…………………………………………….. 44

Figure 9 Predicted outputs using 3-dynamic models in model validation: Experiment

input signal (Dark gray line), experiment output (Gray line) and dynamic model

predicted output (Black line)…………………………………………………………... 51

a. First dataset, output of dynamic model with 2 time delays in discrete time domain.

b. Second dataset, output of dynamic model with 2 time delays in discrete time

domain.

c. First dataset, output of dynamic model with 2 time delays in continuous time

domain.

d. Second dataset, output of dynamic model with 2 time delays in continuous time

domain.

e. First dataset, output of dynamic model with 1 time delay in discrete time domain.

f. Second dataset, output of dynamic model with 1 time delay in discrete time

domain.

Figure 10 Predicted outputs using 3-dynamic models: Experiment input signal (Dark gray

line), experiment output (Gray line) and dynamic model predicted output (Black

line)…………………………………………………………………………………… 53

a. Output of dynamic model with 2 time delays in discrete time domain.

b. Output of dynamic model with 2 time delays in continuous time domain.

c. Output of dynamic model with 1 time delay in discrete time domain.

Figure 11 Predicted results using 3-dynamic models at different impeller speeds…… 55

a. f (channeling) versus impeller speed (rpm).

xiii

b. versus impeller speed (rpm).

Figure 12 Predicted results using 3-dynamic models at different mass concentrations. 61

a. f (channeling) versus impeller speed (rpm) for 0.5% mass concentration.

b. versus impeller speed (rpm) for 0.5% mass concentration.

c. f (channeling) versus impeller speed (rpm) for 1.0% mass concentration.

d. versus impeller speed (rpm) for 1.0% mass concentration.

e. f (channeling) versus impeller speed (rpm) for 1.5% mass concentrations.

f. versus impeller speed (rpm) for 1.5% mass concentration.

Figure 13 Predicted results using 3-dynamic models at 896 L/h flow rates….……….. 64

a. f (channeling) versus impeller speed (rpm) for fluid flow rate 896 L/h.

b. versus impeller speed (rpm) for fluid flow rate 896 L/h.

Figure 14 Predicted results using 3-dynamic models for different output locations…... 67

a. f (channeling) versus impeller speed (rpm) for Configuration-1.

b. versus impeller speed (rpm) for Configuration-1.

c. f (channeling) versus impeller speed (rpm) for Configuration-2.

d. versus impeller speed (rpm) for Configuration-2.

Figure 15 sensitivity analysis plots…………………………………………………….. 70

a. Root mean square (rms) error versus .

b. Root mean square (rms) error versus .

c. Root mean square (rms) error versus .

d. Root mean square (rms) error versus .

LIST OF TABLE:

Table 1 Variables used in experimental work………………………………………… 46

Table 2 Predicted parameters by dynamic models for first set of data..……………… 49

Table 3 Comparison of time taken for simulation by computer……………………… 56

Table 4 Root mean square errors obtained from the dynamic model

with 1 time delay……………………………………………………………... 57

xiv

NOMENCLATURE

The i-th discrete-time model parameter

Positive size-variation factor for

Positive fraction to reduce in equations (56, 57, 58)


Domain of the -th parameter

Minimum value of any

Fraction of short-circuiting, %

-th string evaluation

Average evaluation of all strings in population

Volumetric flow rates, L/h

Transfer function of short-circuiting zone

Transfer function of mixing zone

Transfer function of mixing vessel

Objective function given by equation (51)

Optimal of

Augmented objective function given by equation (56, 57, 58)

Vector of partial derivatives of with respect to

Impeller speed, rpm

Number of bits for -th parameter

Number of crossover sites for each optimization parameter

Number of genetic generations

Size of population of

Number of experiment sample

Number of optimization parameter

Probability of crossover

Probability of mutation

Fluid flow rate, L/h

Backward shift operator

xv

Penalty term in equation (56, 57, 58)

Fraction of recirculation, %

Pseudo-random number

Laplace transform variable

Time, s

Sample time, s

The i-th time delay, s

Input signal, mS (MicroSiemens)

at i-th sampling instant, mS

Volume in short-circuiting zone, m3

Volume in mixing zone, m3

Fully mixed volume in mixing vessel, m3

Total volume fluid in mixing vessel, m3

Output signal, mS


Experimental output signal, mS


Vector of optimization parameters

Optimal of

Vector of mean value of

Vector of deviation of from

in binary coding

GREEK LETTERS

The i-th time constant, s


A small positive number

1

CHAPTER 1

INTRODUCTION

1.1 Background

Continuous-flow mixing is a vital component to many processes including

polymerization, fermentation, wastewater treatment, and pulp and paper

manufacturing. its demonstrated ability to improve product uniformity and minimize

shutdown, loading, and unloading times. Continuous mixing is considered as an

efficient alternative to a batch mixing operation.

Understanding continuous mixing mechanisms is crucial especially when the

fluids are non-Newtonian. A wide range of working fluids in industrial mixers are

non-Newtonian mostly pseudoplastic or shear thinning fluids with yield stress.

Important non-Newtonian fluids such as pulp suspensions, food substances like

margarine and ketchup, paints, cement, and certain polymer and biopolymer solutions

are pseudoplastic with yield stress. These non-Newtonian fluids create considerable

deviations from ideal mixing because continuous mixing of non-Newtonian fluids

exhibit non idealities such as channeling, recirculation, and stagnant zones.

It is very important to characterize non idealities present during the continuous

mixing of non-Newtonian fluids. Different dynamic models were developed in order

to represent the dynamics of continuous mixing for non-Newtonian fluids. In this

study, three dynamic models were utilizes to characterize the continuous mixing of

non-Newtonian fluids: (1) a dynamic model with 2 time delays in discrete time

domain, (2) a dynamic model with 2 time delays in continuous time domain, and (3) a

simplified dynamic model with 1 time delay in discrete time domain. Hybrid genetic

algorithm was used in order to identify dynamic model parameters.

2

1.2 Thesis objective and contribution

1.2.1 Problem statement

The continuous mixing of non-Newtonian fluids exhibits non idealities such as

channeling, recirculation, and stagnant zones. Presence of non idealities is one of the

main reasons for poor mixing performance of continuous mixer. It is important to

identify the non idealities present during the continuous mixing of non-Newtonian

fluids. The dynamic models which represent the dynamics of continuous mixing have

two different concepts: (1) the dynamic models which have 2 time delay terms and

(2) the dynamic model which has 1 time delay term. It is important to observe the

effect of time delays in characterization of continuous mixing of non-Newtonian

fluids.

1.2.2 Dynamic models

In this study, three dynamic models were utilized for characterization of continuous

mixing of non-Newtonian fluids:

1. Ein-Mozaffari‟s dynamic model (Ein-Mozaffari et al., 2004)

2. Patel‟s dynamic model (Patel et al., 2007)

3. Soltanzadeh‟s dynamic model (Soltanzadeh et al., 2008)

Ein-Mozaffari‟s dynamic model and Patel‟s dynamic model both have 2 time delays

term while Soltanzadeh‟s dynamic model has 1 time delay term.

1.2.3 Hybrid genetic algorithm

Genetic algorithms (GAs) are global optimization techniques that are particularly well-

suited for highly nonlinear functions. GAs can handle noisy, discontinuous functions.

GAs can rapidly locate the region in which the global optimum exists but they take

relatively long time to locate the exact global optimum in the region of convergence. A

combination of GA and a local search method can speed up the search to locate the

exact global optimum. In such a hybrid, applying a local search to the solutions that are

guided by a GA to the most promising region can accelerate convergence to the global

optimum. The time needed to reach the global optimum can be further reduced if local

3

search methods and local knowledge are used to accelerate to locating the most

promising search region in addition to locating the global optimum starting within its

basin of attraction. Hybrid genetic algorithm can overcome genetic drift that arises as a

result of finite population size. In this study “hybrid genetic algorithm” was utilized for

identification of the dynamic models parameters, which uniquely integrating genetic

algorithms with gradient search (a local search method).

1.3 Thesis layout

This thesis is organized in the following fashion: Chapter 2 includes an introduction of

mixing, rheology of non-Newtonian fluids, and literature review on the continuous

mixing of pseudoplastic fluid with yield stress. Chapter 3 introduces different dynamic

models which used in characterization of the continuous mixing of non-Newtonian fluids.

Chapter 4 details the system model that is the main focus of the identification procedure.

Once the system has been modeled, the identification technique is introduced. Based on

this objective function, the procedures to identify dynamic model parameters using

hybrid genetic algorithm is formulated. A sensitivity analysis method is introduced.

Chapter 5 includes genetic algorithms strength, weakness, and importance to hybridize

the genetic algorithms. Chapter 6 includes overview of experimental work and rheology

of xanthan gum. Chapter 7 includes simulation results and discussions. Chapter 8 details

the conclusion, overall result of this research and possible future research in this topic.

5

CHAPTER 2

MIXING

2.1 Introduction

Mixing plays an important role in different industries such as food, pharmaceutical,

paper, plastic, ceramics, and polymers. Mixing is a unit operation that involves

manipulating a heterogeneous physical system with the intent to make it more

homogeneous. There are two modes of mixing at the industrial level:

1. Batch Mixing Operation

2. Continuous Mixing Operation.

2.1.1 Batch Mixing Operation

A batch mixing process typically consists of three sequential steps: loading mix

components, mixing, and discharge of the mixed product. In a batch mixing process all

ingredients are loaded into a mixer and mixed for duration until they are homogenously

mixed. The retention time in a batch mixer is normally arrived at based on trials wherein

the time required for achieving the desired level of product homogeneity is established.

Mixing cycle times can range from a few seconds with high intensity units to many hours

where an additional processing like heating or cooling may be involved. The resulting

mix is then discharged out of the mixing vessel. (Tekchandaney, 2009)

The total batch time is the time required for charging the material into the mixer, the

mixing cycle time and the discharge time. For applications where product contamination

between successive batches is not permitted, the time required for cleaning the mixer

should also be added to the total batch time.

2.1.1.1 Applications of Batch Mixing

Batch mixing is preferred for applications where:

Many formulations are produced on the same production line.

Ingredient properties change over time.

6

Production quantities are small.

Strict control of mix composition is required.

2.1.1.2 Advantages of Batch Mixing

The advantages of the batch mixing operation are as follows:

Precise control of mix quality.

Batch traceability

Lower installed and operating costs for small to medium capacities compared to

continuous mixing

Flexibility of production.

Easy cleaning, lower cleaning costs when product changes are frequent.

2.1.1.3 Disadvantages and Limitations of Batch Mixing

The disadvantages of batch mixing are as follows:

Batch mixing is uneconomical when large quantities of material are to be mixed.

Batch mixing is requiring more labor compared to continuous mixing.

2.1.2 Continuous Mixing Operation

Continuous mixing is used to mix ingredients continuously in a mixer in a single pass. In

a continuous mixing process, the loading, mixing, and discharge steps occur continuously

and simultaneously.

The materials to be mixed are continuously charged into the mixer as per the

formulation. The process of charging the materials in a continuous mixer is extremely

critical and can significantly affect the quality of the final mix. The time taken by the

material to travel from the feeding point to the discharge point is known as the retention

time of the material in the mixer. Material retention time is not uniform in continuous

mixing and can be directly affected by mixer speed, feed rate, mixer geometry,

inlet/outlet location, and the design of mixer internals. Material is continuously

7

discharged at a constant rate which is generally termed as the capacity of the continuous

mixer. (Tekchandaney, 2009)

2.1.2.1 Applications of Continuous Mixing

Continuous mixing is preferred for applications where:

Large quantities of a single product are to be mixed.

Requiring high production rate.

Strict batch integrity is not critical.

Smoothing out batch product variations is required.

2.1.2.2 Advantages of Continuous Mixing

The advantages of the continuous mixing operation are as follows:

High Capacity - Compared to batch type mixers, continuous mixers of smaller

volumes and power can be used to produce large quantities of uniform mix.

Hence for a given capacity they are more compact than batch mixers.

Lower Mixing Time - The mixing time in continuous mixers is lower than in

batch mixers.

Consistent Mixing Performance – With proper feeding arrangements, online

instrumentation and operation controls, a consistent mixing performance and

uniform product quality can be achieved.

Suitability for Automatic Control - Operation of continuous mixers can be

automated using online monitoring and measuring instruments.

Lower Cost of Mixers - Continuous mixers tend to be cheaper than the

equivalent batch mixers because they are compact and require less space.

However the cost of feeders for metering the product into the mixer,

instrumentation and control may result in a higher overall cost of the system.

Minimum Labor – Since material feeding and discharging processes are

automated, minimal labor is required for continuous mixing.

8

2.1.2.3 Disadvantages and Limitations of Continuous Mixing

Lack of Flexibility – Continuous mixing systems are designed for a particular

application and cannot be easily tailored to mix different formulations. Even if a

new ingredient is to be introduced, it calls for a change in the protocol, and the

system has to be recalibrated.

Component Limitations – When a large number of ingredients are to be added,

continuous mixers have limitations with respect to mixing uniformity when

compared to batch mixers.

Higher Overall Maintenance Cost – Continuous mixers heavily depend on

feeders, instrumentation and online control systems. Failure, malfunction in any

one component can lead to complete stoppage. Hence, overall maintenance costs

for continuous mixers are higher compared to batch mixers.

Calibration and Checking – The feeding devices in a continuous mixing require

careful calibration and frequent checking for accuracy.

Critical Applications – Continuous mixers are not suited for critical applications

where product formulations need to be exact. Batch mixers are better suited to

processes that require a very tight product formulation and uniform composition.

2.2 Continuous mixing of non-Newtonian fluids

2.2.1 Introduction

Continuous-flow mixing is widely used in many processes such as polymerization,

fermentation, wastewater treatment, and pulp and paper manufacturing because of its

demonstrated ability to improve product uniformity and minimize shutdown, loading, and

unloading times. Continuous mixing of non-Newtonian fluids is difficult to characterize

due to the complex rheology of non-Newtonian fluids.

Understanding continuous mixing mechanisms is crucial especially when the fluids

are non-Newtonian. A wide range of working fluids in industrial mixers are non-

Newtonian mostly pseudoplastic or shear thinning fluids with yield stress. Important non-

Newtonian fluids such as pulp suspensions, food substances like margarine and ketchup,

9

paints, cement, and certain polymer and biopolymer solutions are pseudoplastic with

yield stress. The mixing of non-Newtonian fluids is often described as an “art” rather

than a “science”, depending upon empirical correlations, working experience, and

intuitive knowledge (Sue and Holland, 1968).

2.2.2 Rheology of non-Newtonian Fluids

Rheology studies the relation between force and deformation in materials. The

relationship between stresses acting at a point in a fluid and deformations occurring as a

result of their action is called rheological equation of state or constitutive equation

(Malkin, 1994).

Viscosity is usually defined as the ratio of shear stress (τ) to shear rate ( ). Fluids

having a constant viscosity for any shear rate are called Newtonian and their viscosity is

called Newtonian viscosity. If the ratio of shear stress to shear rate is not constant, the

fluid is called non-Newtonian, and their viscosity is called apparent or non-Newtonian

viscosity (Morrison, 2001).

Non-Newtonian fluids can be grouped into three different classes (Chhabra and

Richardson, 1999) as indicated in Figure 1.

2.2.2.1 Time-Independent Fluids

Time-independent fluids are the most common type of non-Newtonian fluids. At any

time, shear rate at any point in these fluids is determined only by the value of shear stress

(Tanner, 2000). Under this category, two types of non-Newtonian fluids can be

distinguished: fluids with yield stress and fluids without yield stress.

10

Figure 1: Categories of non-Newtonian fluids

Some non-Newtonian fluids show little or no deformation up to a certain level of stress.

Such fluids behave like a solid below the level of stress and like a fluid above it. The

stress at which such fluids start to flow is called yield stress, resulting in fluids often

referred to as yield stress fluids (Barnes, 2000). The internal structure of yield stress

fluids is capable of preventing movement below yield stress. However, the internal

structure reforms to allow the fluid to move when shear stress exceeds yield stress

(Macosko, 1994). Examples of fluids possessing yield stress can be found in plastic

melts, coal, cement, margarine and shortenings, greases, chocolate mixtures, toothpaste,

soap and detergent slurries, and paper and pulp suspension. Non-Newtonian fluids

without yield stress, conversely, can flow at any stress. Examples of non-Newtonian

fluids without yield stress include starch suspensions, fruit juice concentrates and

printer's ink.

Non-Newtonian fluids with and without yield stress can be further divided into

pseudoplastic and dilatants fluids. While pseudoplastic fluids thin at high shear rates,

dilatants fluids thicken at high shear rates. Most of the non-Newtonian fluids encountered

11

in industries are pseudoplastic or shear thinning. These fluids characterized by apparent

viscosity which decreases with increasing shear rate. Examples of pseudoplastic fluids

are rubber solutions, adhesives, soap, detergents, paints, and paper and pulp suspension.

Dilatants fluids characterized by apparent viscosity increases with increasing shear rate;

thus these fluids are also called shear-thickening. Examples of dilatants fluids are corn

flour/sugar solutions, and many pigment dispersions containing high concentration of

suspended solids such as mica and powdered quartz.

2.2.2.2 Time-Dependent Fluids

Shear rate in time-dependent fluids is a function of both magnitude and the duration of

shear stress application to the fluid. Thixotropic and rheopectic or anti-thixoptropic fluids

are the most common kinds of time dependent non-Newtonian fluids.

In thixotropy, the longer the fluid is subjected to shear stress, the lower its viscosity.

However, in rheopecty, the longer the fluid is subjected to shear stress, the higher its

viscosity. Many gels, paints, and printing inks are thixotropic, exhibiting a stable form at

rest but becoming fluid when agitated (subjected to shear stress). Some lubricants and

gypsum suspension thicken or solidify when shaken and hence are considered rheopectic

fluids (Barnes, 2000).

2.2.2.3 Viscoelastic Fluids

Viscoelastic fluids exhibit the characteristic of both fluids and elastic solids, showing

partial elastic recovery after deformation (after applying shear stress).

In a purely elastic solid the shear stress corresponding to a given shear rate is

independent of time, whereas for viscoelastic substances the shear stress will gradually

dissipate with time. In contrast to purely viscous liquids, viscoelastic fluids flow when

subjected to shear stress but part of their deformation is gradually recovered upon

removal of shear stress (Macosko, 1994). Examples of viscoelastic fluids include

bituminous, flour dough, and some polymer and polymer melts such as nylon.

http://en.wikipedia.org/wiki/Shear_stress

http://en.wikipedia.org/wiki/Gel

http://en.wikipedia.org/wiki/Lubricant

12

2.2.3 Literature review on continuous mixing of pseudoplastic with yield stress

fluids

Little information is available on continuous mixing of non-Newtonian fluids. It is

important to investigate the dynamic behavior of continuous mixing. Traditionally

continuous mixers have been designed assuming they are ideally mixed, with the

dynamics represented by a first order transfer function. In order to understand continuous

mixing mechanisms of non-Newtonian fluids, Ein-Mozaffari et al. (2002, 2005) studied

the dynamics of a rectangular agitated pulp stock chest under controlled conditions of

impeller speed, pulp suspension concentration, input pulp suspension flow-rate and vessel

input and output locations. They attributed the deviations from ideal mixing to a number

of factors including short-circuiting, recirculation and the presence of dead zones in the

chest. Such zones can arise from the interaction between the circulation patterns

generated by the impellers, the suspension flow through the vessel and chest geometry

(Ein-Mozaffari et al., 2004).

Dynamic tests conducted on a rectangular pilot stock chest by Ein-Mozaffari et al.

(2003) showed that the extent of non-ideal flow can be significant. The percentage of

channeling was as high as 90% of pulp feed at high input flow-rates and low impeller

speed. Ein-Mozaffari et al. (2004) developed a dynamic model based on physical in-

terpretation of the flow field and used it to characterize the non-idealities present in the

continuous mixing. This dynamic model (Ein-Mozaffari et al., 2004) comprised two

separate time delays one for a short-circuiting zone (channeling) and one for a mixing

zone.

Ford et al. (2006) used a computational-fluid-dynamics (CFD) model of a rectangular

pulp-stock mixing chest and reported that dynamic performance of pulp chest was far

from ideal, with a significant extent of non-idealities. They showed that the dynamic

model proposed by Ein-Mozaffari et al. (2004) capture the mixing dynamic of the scaled

model chest fairly well. Soltanzadeh et al. (2008) used a simplified version of the model

developed by Ein-Mozaffari et al. (2004) and tried to describe the mixer dynamic and

flow non-idealities for a rectangular pulp chest. They used standard PEM (prediction-

13

error minimization) function in MATLAB identification toolbox to predict the extent of

non-ideal flows. They considered that the time delays for both mixing and short-

circuiting zones were equal.

Saeed et al. (2008) analyzed the performance of the continuous flow mixing in a

cylindrical vessel. The fluid used in the experiment was xanthan gum solution which is a

pseudoplastic fluid with yield stress. They showed that the performance of a continuous-

flow mixer can be improved by increasing the impeller speed, decreasing the flow rate,

and decreasing the yield stress (by reducing the solution mass concentration), relocating

the input/output location and proper type of impeller. CFD simulation was done in order

to match the parameters of the dynamic model proposed by Ein-Mozaffari et al. (2004)

with the experimental ones.

It is known that accurate modeling, precise parameterization and identification are

very important to represent the dynamic of continuous mixing. Identification of dynamic

model parameters has long been the topic of research interest in the past (Johansson,

1994; Johansson et al., 1999; Soderstrom et al., 1997; Whitfield and Messali., 1987), The

identification of systems with time delays have been reported (Sung and Lee, 2001;

Wang et al., 2001) but these schemes are not generic enough to handle Ein-Mozaffari et

al. (2004) dynamic model. Kammer et al. (2005) developed a numerical method for

estimating model parameters in the model developed by Ein-Mozaffari et al. (2004). In

this method, the estimation of the time delays was performed independently from the

estimation of the remaining parameters of the model. The authors proposed two distinct

stages for the identification: an efficient but less accurate search (Least Squares

Minimization) for the optimal delays, followed by an accurate search (Sequential

Quadratic Programming) for the whole set of parameters. Although this mechanism is not

guaranteed to converge to the global minimum, a Monte Carlo simulation showed very

encouraging results.

Upreti and Ein-Mozaffari (2006) presented a technique for the identification of non-

ideal flows. This technique provides a robust, good quality solution independent of

14

starting points, and auxiliary conditions. The technique is based on genetic algorithm

(Holland, 1975) and they obtained better results in comparison to iterative dynamic

programming as well as Sequential Quadratic Programming. Upreti and Ein-Mozaffari

(2006) determined the optimal parameters of the pulp chest model using this technique.

The technique “hybrid genetic algorithm” was developed by uniquely integrating genetic

algorithms with gradient search. The hybrid algorithm identified the optimum parameters

with high accuracy.

Patel et al. (2007) carried out simulation and identified mixing parameters of agitated

pulp stock chests in a continuous time domain. The differential algebraic model was

developed in order to represent the continuous mixing dynamic in continuous time

domain. The differential algebraic model obviates the restrictions (zero order hold)

imposed by the discrete time approaches. Hybrid genetic algorithm was utilized along

with the differential algebraic model. They characterized the dynamic of agitated pulp

stock chest at a higher sampling time. The differential algebraic model considered two

separate time delays one for a short-circuiting zone (channeling) and one for a mixing

zone. Patel et al. (2010) characterized the agitated pulp chest and obtained superior

characterizations compared to those yielded by the discrete time domain approach.

15

CHAPTER 3

MATHEMATICAL MODEL

3.1 Introduction

Eykhoff (1974) defined a mathematical model as „a representation of the essential

aspects of an existing system (or a system to be constructed) which presents

knowledge of that system in usable form.

A mathematical model usually describes a system by a set of variables and a set

of equations that establish relationships between the variables. The value of the

variables can be practically anything such as real or integer numbers, Boolean values

or strings. The variables represent some properties of the system, for example,

measured system outputs often in term of signals, timing data, counters, and event

occurrence (yes/no). The actual model is the set of functions that describe the relations

between the different variables.

Classification of mathematical models

Most mathematical models can be classified in the following ways (Kapur, 1998):

(1) Linear vs. nonlinear: mathematical models are usually composed by variables, which

are abstractions of quantities of interest in the described systems, and operators that

act on these variables, which can be algebraic operators, function, differential

operators, etc. if all the operators in a mathematical model present linearity, the

resulting mathematical model is defined as linear. A model is considered to be

nonlinear otherwise.

(2) Deterministic vs. probabilistic (stochastic): A deterministic model is one in which

every set of variable states is uniquely determined by parameters in the model and by

sets of previous states of these variables. Therefore, deterministic models perform the

same way for a given set of initial conditions. Conversely, in a stochastic model,

16

randomness is present, and variable states are not described by unique values but

rather by probability distributions.

(3) Static vs. dynamic: A static model does not account for the element of time, while a

dynamic model does. Dynamic models typically are represented with difference

equations or differential equations.

(4) Discrete vs. continuous: A discrete model in which system jumps from one state to

the next at fixed interval or time steps. A continuous model in which system changes

continuously over time. The continuous model is more amenable to algebraic

manipulation.

(5) Non-parametric vs. parametric: Non-parametric model in which the model structure

is not specified a prior but is instead determined from data. The term Non-parametric

is not meant to imply that such models completely lack parameters but that the number

and nature of the parameters are flexible and not fixed in advance. The parametric

model has specified structure.

3.2 Dynamic models for continuous mixing of non-Newtonian fluids

3.2.1 Ein-Mozaffari’s dynamic model

As mentioned earlier, different models have been developed to describe the dynamics of

continuous fluid mixing.

Figure 2 shows block diagram of the dynamic model proposed by Ein-Mozaffari et al.

(2004) for continuous mixing, which has two distinct zones

1. A mixing zone consisting of a first order transfer function with time delay and

feedback for recirculation.

2. A short-circuiting zone consisting of a first order transfer function plus time delay.

17

u y

Figure 2: Block diagram of Ein-Mozaffari‟s dynamic model

On the basis of this dynamic model a fraction of the incoming feed-stream, f, by-passes

the mixing zone and moves directly to the exit of the mixing vessel while the remaining

fraction (1 – f ) enters the mixing zone around the impeller. The primary mixing occurs in

the mixing zone and poor mixing could also occur in the short-circuiting zone. Mixing in

both zones was represented by a first order transfer function ( and ). and are

the time delays for short-circuiting and mixing zone respectively. Time delay is the time

interval between the changes in input variable and time the output variable start to

respond. and are the time constants for short-circuiting and mixing zone

respectively. Time constant is the time required for the output variable to reach 63.2% of

the total change when the input variable change in a step fashion. u and y are input and

output signals, respectively, they are either the pulp fiber mass concentration, in

industrial settings, or suspension conductivity, in the scale model. The combine transfer

function of mixing vessel for the dynamic model proposed by Ein-Mozaffari et al. (2004)

in continuous time domain is given by:

f

1- f

R

1- R

18

The model reduces to a first-order transfer function, representing an ideally mixed chest,

when f = 0 and R = 0, with the time constant of a mixing zone approaches the dominant

time constant.

3.2.2 Patel’s dynamic model

Figure 3 shows the dynamic model proposed by Patel et al. (2007) for the continuous

mixing process with the same fundamental used by Ein-Mozaffari et al. (2004). This

model also includes two separate time delays T1 and T2 for short-circuiting and mixing

zones, respectively. The zero order hold which was used for discrete time domain

approach is not considered in continuous time domain for identification of dynamic

model parameter.

fF, u fF, y1

Input Output

F, u (1 - f )F, u F2, yj F2, y2 (1 - R )F2, y2 F, y

R F2, y2

Figure 3: Block diagram of Patel‟s dynamic model

The output signal at a time [y (t)] is dependent on the intermediate output of short-

circuiting zone ( ) and mixing zone ( ). F is the volumetric flow rate, V1 and V2 are the

volumes of the short-circuiting and mixing zone, u and y are the input and output signals

of the mixing vessel. With time ( t ) as the independent variable, the mass balances for

two zones are given by following differential equations:

Short-circuiting zone Of volume

Short-circuiting zone Of volume

19

With R as the fraction of pulp re-circulated within the mixing zone,

The system output is as follow:

where and are both zero for negative times.

The initial condition for equation (2) and (3) are

where is the experimental output specified at the initial steady state.

3.2.3 Soltanzadeh’s dynamic model

Figure 4 shows Soltanzadeh et al. (2008) model a simplified version of Ein-Mozaffari et

al. (2004) dynamic model, which has two distinct zones

1. Mixing zone

2. Short-circuiting zone.

The model interpreted that the fluid stream flows in a plug flow from inlet to the mixing

zone where fraction f of fluid separates and proceeds to the outlet through the short-

circuiting zone. The fraction 1 – f passes through the mixing zone and proceeds to the

outlet of the mixing vessel.

20

f

u y

1- f

Figure 4: Block diagram of Soltanzadeh‟s dynamic model

The time delays for two zones are equal and merged so that only one time delay term is

considered (T1 = T2 = T). Thus fluid takes equal time to pass from inlet to outlet of the

mixing vessel for both zones. Insignificant mixing happens in the short-circuiting zone

which is modeled as a first-order transfer function with time constant . Significant

mixing is expected to take place in the mixing zone and is modeled as the transfer

function with a time constant, . The transfer function for both short-circuiting and

mixing zone G1 and G2 are first order with the same time delay. The combined transfer

function of mixing vessel (system) for dynamic model proposed by Soltanzadeh et al.

(2008) in continuous time domain is given by:

21

CHAPTER 4

SYSTEM IDENTIFICATION

4.1 Introduction

Generally speaking, the system identification method is a “black box” process for

estimating the dynamic system under study using simultaneously measured input and

output data. The procedure is typically iterative and consists of three basic steps: (a)

data generation/collection, (b) model determination and (c) model validation (see

Figure 5)

Prior knowledge

Model determination

Not ok

Revise

Figure 5: Flow diagram of system identification (Xiao, 2009)

Data generation /

Collection

Candidate

models

Criteria

of fit

Model estimation

Model validation

22

For the data generation/collection step, does not only involve choosing the signals to

be measured and by what means but also designing the experiments so that the measured

signals are sufficiently informative for the subsequent procedures. More significantly, for

nonparametric models, which do not assume any particular structure, the signals should

contain all the frequencies of the system under study. On the other hand, for parametric

models, which assume a particular structure, the input signals should contain at least as

many frequency components as the number of parameters characterizing the system.

Model determination is the step to choose a model that “best” couples the input-output

data from a pool of candidates. Although nonparametric models are attractive in the sense

that they do not assume any particular mathematical structure and are relatively easy to

estimate, they are only applicable to systems operating in open-loop conditions, which

are, unfortunately, not the case in continuous mixing mechanisms. Therefore, parametric

models are more often used for analyzing continuous mixing mechanisms. Another key

question that must be addressed in establishing a set of candidate models is: can the

system under study be represented with linear and time-invariant (LTI) model? The

complex mixing system is generally nonlinear and time varying, however, LTI models

may be nearly valid when the data are collected during short time periods of stable,

unchanging experimental conditions.

After an optimal model is chosen, its parameters are usually determined by

minimizing the variance of the unobserved stochastic disturbance. The hybrid genetic

algorithm approach is appropriate when the unobserved disturbance is normally

distributed.

Once the model is determined, a final step remains as to demonstrate the validity of

the model for its intended purpose. The most obvious approach for model validation

would be to obtain an independent measure of the system dynamics (without the use of

system identification) against which the estimated model may be compared. Thus model

validation is based on the way in which the model is used, prior information on the

system, the fitness of model to real data, etc. For example, if we identify the transfer

23

function of a system, the quality of an identified model is evaluated based on the step

response and/or the pole-zero configurations. Further more if the ultimate goal is to

design a control system, then we must evaluate control performance of a system designed

by the identified model. If the performance is not satisfactory, we must go back to some

earlier stage of system identification as shown in the Figure 5.

Model identification comprised two experiments. In the first experiment, the input

signal was a rectangular pulse which allowed the estimation of an approximate model to

design the excitation for the second experiment (Kammer et al., 2005). The excitation

energy for the second experiment was concentrated at frequencies where the Bode plot is

sensitive to parameter variations (Ljung, 1999). Therefore, the frequency-modulated

random binary input signal was designed for this purpose.

In this study three different dynamic model were utilized in order to describe the

dynamics of continuous mixing. To represent dynamics of continuous mixing, the

dynamic model proposed by Ein-Mozaffari et al. (2004) has total six parameters, the

dynamic model proposed by Patel et al. (2008) has six parameters, and the dynamic

model proposed by soltanzadeh et al. (2009) has five parameters. The recirculation was

not observed in lab scale continuous mixer so it was neglected in all three dynamic model

parameter estimation.

4.2 Estimation of parameters via hybrid genetic algorithm

Extensive and profound studies have been performed in the area of parameter estimation.

Generally, conventional techniques, such as Least Square Estimator (LSE), Maximum

Likelihood Estimator (MLE), Bayes Estimator (BE), instrumental Variable Methods

(IVM), and so on, can deliver adequate identification results. However, the conventional

approaches often depend on the calculation of gradients (the first or higher order

derivatives of the cost function with respect to the unknown parameters) greatly. For the

non-differentiable cost functions, those methods will not be able to locate the optimal

estimates of the unknown parameters. Even for the differentiable cost functions, if they

24

have multiple local minima, the gradient-based methods suffer from being trapped at one

of the local minima. That makes them very sensitive to the initial guesses of the estimated

parameters. Furthermore, although the conventional techniques very often provide good

results with respect to convergence and error minimization, they also are limited in their

ability to handle the presence of noise (Sheta and De Jong, 1996).

The difficulties in the field of parameter estimation naturally lead to the exploration

of other search techniques for parameter estimation of nonlinear systems. In particular,

global optimization techniques have been introduced into the field of parameter

estimation. Genetic Algorithms (GAs) are population-based global optimization

techniques that emulate natural genetic operators. Because they simultaneously evaluate

many points in the parameter space, they are very likely to converge toward the global

solution. They require no gradient information or continuity of the searching space.

Therefore, application of GAs in the field of parameter estimation and system

identification has received a lot of attention over the last two decades.

Although GAs have appealing properties and have been applied successfully in the

field of parameter estimation, they could be very computationally inefficient, especially

for complex nonlinear dynamic systems with multiple unknown model parameters. As

mentioned in previous discussion, GAs for solving optimization problem consist of a

sequence of computational steps, which asymptotically converge at an optimal solution.

Differing from the most classical point-to-point optimization techniques, such as the

Newton-Raphson method, GAs performs a multiple directional search by maintaining a

population of potential solutions. Each iteration has a large number of points in the

parameter space are evaluated. While the population to population approach makes the

search escape from local optima, the price is paid by increasing computation time

dramatically.

In this study, a hybrid genetic algorithm-based parameter estimation strategy is

proposed to improve the computational efficiency of GAs yet without losing their

advantages. It starts with a GA to search for „good‟ starting points globally through the

25

possible solution areas, and then applies a gradient-based optimization approach to find

the solution in the likelihood sense. Instead of selecting starting points arbitrarily, the

hybrid genetic algorithm provides a systematic search procedure, such that MLE is more

likely to achieve the global minimum. In addition, the hybrid genetic algorithm is more

efficient than pure GAs because the gradient-based techniques converge along the

deepest descending direction. Further description of the GAs and hybrid genetic

algorithm is given in Chapter 5.

There were two different approaches used for parameter estimation of dynamic model

which represent dynamic of the continuous mixing system, along with the hybrid genetic

algorithm.

1. Discrete time domain approach

2. Continuous time domain approach

4.2.1 Discrete Time Domain Approach

Since the data are measured at fixed time intervals, the continuous model is initially

transferred in to a simple discrete-time model (Kammer et al. 2005). By applying zero-

order hold to the transfer function (Equation 1), the equivalent discrete time transfer

function is given by Equation 9.

26

The Z-transformation contains the value at the sampling instants. These values depend

on the sampling time also the transform does not contain information on the function

for the negative time. In this approach, it was assumed that the time delays were multiple

of the sampling time. Equation 1 can also discretized by modified Z- transform but this

would add more terms to the Equation 9. Since the sampling time is small enough to

describe time delays with significant accuracy, the consideration of fractional time delays

is not necessary. The zero-order hold equivalent to each of the transfer functions

(Seborg et al. 1989) in above equations is

A zero-order hold simply holds the value of the input between samples without

accumulation and it describe the effect of converting a continuous-time-signal to a

discrete-time signal by holding each sample value for one sample interval.

The dynamic model with 2 time delays (Ein-Mozaffari et al. 2004) in discrete time

domain was given by Equations from (9) to (14). It was further augmented, for

constraints (from Equation 16 to Equation 18) and penalty function (Equation 56). The

output from the dynamic model with 2 time delays in discrete time domain (Ein-

Mozaffari‟s model) was given by Equation 19. Related derivatives (from Equation 20 to

Equation 26) were derived and solved using hybrid genetic algorithm.

The output signal

27

Derivatives for gradient search method

The dynamic model with 1 time delay (Soltanzadeh et al. 2008) in discrete time

domain was given by Equations from (27) to (32). This model was further augmented, for

constraints (from Equation 33 to Equation. 35) and penalty function (Equation 57). The

output from the dynamic model with 1 time delay in discrete time domain

28

(Solatanzadeh‟s model) was given by Equation 36. Related derivatives (from Equation 37

to Equation 43) were derived and solved using hybrid genetic algorithm.

Transfer function

The output signal

Derivatives for gradient search method

29

4.2.2 Continuous Time Domain Approach

In recent years, with the advent of inexpensive computing power, much work has been

done for study of discrete time domain approaches. Today, system parameters are

routinely estimated through many methods (standard least-square, extended least-squares,

generalized least-squares, and maximum likelihood methods). However, many

applications, especially those in the process industries, are inherently continuous time

systems. In many of these cases, continuous-time parameter estimates are more

meaningful when characterizing system dynamics. These parameters can be indirectly

30

estimated by transforming the discrete-time transfer function into the continuous-time

domain.

The dynamic model with 2 time delay (Patel et al. 2008) in continuous time domain

was given by equations from (2) to (6). This model was further augmented, for

constraints (from Equation 44 to Equation 46) and penalty function (Equation 58). The

system output in continuous time domain, for patel‟s model is given by Equation 6.

Related partial derivatives (from Equation 47 to Equation 50) were derived and solved

using hybrid genetic algorithm. For continuous time domain approach, fifth-order Runge-

Kutta Fehlberg method with Cash-Karp parameters, and adaptive step-size control (Press

et al., 2002) in conjunction with the genetic operations, were used.

Partial derivatives of for continuous time domain approach

31

4.3 Sensitivity analysis

Engineering and scientific phenomena are often studied with the aid of mathematical

models designed to simulate complex physical processes. In the process industry,

modeling the continuous mixing of non-Newtonian fluid is important to know mixing

dynamics in the mixers. One of the steps in the model development is the determination

of the parameters, which are the most influential on model results. A “sensitivity

analysis" of these parameters is not only critical to model validation but also serves to

guide future research. Sensitivity analysis is used to determine how “sensitive” a model is

to changes in the value of the model parameters. Parameter sensitivity is usually

performed as a series of tests in which the modeler sets different parameter values to see

how a change in the parameter causes a change in the dynamic model response. By

showing how the model behavior responds to changes in parameter values, sensitivity

analysis is a useful tool in model building as well as in model evaluation.

Sensitivity analysis helps to build confidence in the model by studying the

uncertainties that are often associated with parameters in models. Many parameters in

system dynamics models represent quantities that are very difficult, or even impossible to

measure to a great deal of accuracy in the real world. Also, some parameter values

change in the real world. Therefore, when building a system dynamics model, the

modeler is usually at least somewhat uncertain about the parameter values he/she

chooses. Sensitivity analysis allows him/her to determine what level of accuracy is

necessary for a parameter to make the model sufficiently useful and valid. If the tests

reveal that the model is insensitive, then it may be possible to use an estimate rather than

a value with greater precision. Sensitivity analysis can also indicate which parameter

values are reasonable to use in the model. If the model behaves as expected from real

world observations, it gives some indication that the parameter values reflect, at least in

part, the “real world.” Sensitivity tests help the modeler to understand dynamics of a

system. Experimenting with a wide range of values can offer insights into behavior of a

system in extreme situations. Discovering that the system behavior greatly changes for a

change in a parameter value can identify a leverage point in the model a parameter whose

specific value can significantly influence the behavior mode of the system.

32

Sensitivity analysis (Hamby, 1994) method used in this study was one-at-a-time

analysis. Sensitivity measure was done by increasing and decreasing the parameter value

by 20% from its base class value. Base class value of parameters is identified value of the

parameter by simulation for a specified dataset.

33

CHAPTER 5

GENETIC ALGORITHM

5.1 Introduction

Genetic algorithms (GAs) are adaptive methods which may be used to solve search and

optimization problems. They are based on the genetic processes of biological organisms.

Over many generations, natural populations evolve according to the principles of natural

selection and "survival of the fittest”. By mimicking this process, genetic algorithms are

able to "evolve" solutions to real world problems, if they have been suitably encoded.

The basic principles of GAs were first laid down rigorously by Holland [1975].

GAs work with a population of "individuals", each representing a possible solution

to a given problem. Each individual is assigned a "fitness score" according to how good a

solution to the problem it is. The highly-fit individuals are given opportunities to

"reproduce", by "cross breeding" with other individuals in the population. This produces

new individuals as "offspring", which share some features taken from each "parent". The

least fit members of the population are less likely to get selected for reproduction, and so

"die out".

A whole new population of possible solutions is thus produced by selecting the best

individuals from the current "generation", and mating them to produce a new set of

individuals. This new generation contains a higher proportion of the characteristics

possessed by the good members of the previous generation. In this way, over many

generations, good characteristics are spread throughout the population. By favoring the

mating of the more fit individuals, the most promising areas of the search space are

explored. If the GA has been designed well, the population will converge to an optimal

solution to the problem.

34

5.1 The method

The evaluation function, or objective function, provides a measure of performance with

respect to a particular set of parameters. The fitness function transforms that measure of

performance into an allocation of reproductive opportunities. The evaluation of a string

representing a set of parameters is independent of the evaluation of any other string. The

fitness of that string, however, is always defined with respect to other members of the

current population. In the genetic algorithm, fitness is defined by: where is the

evaluation associated with string and is the average evaluation of all the strings in

the population.

Fitness can also be assigned based on a string's rank in the population or by sampling

methods, such as tournament selection. The execution of the genetic algorithm is a two-

stage process. It starts with the current population. Selection is applied to the current

population to create an intermediate population. Then recombination and mutation are

applied to the intermediate population to create the next population. The process of going

from the current population to the next population constitutes one generation in the

execution of a genetic algorithm. The standard GA is represented in figure 6.

Selection Recombination

(Duplication) (Crossover)

Current Intermediate Next

Generation n Generation n Generation n+1

Figure 6: Standard genetic algorithm

String 1

String 2

String 3

String 4

String 1

String 3

String 3

String 4

Offspring-A(1Χ3)

Offspring-B(1Χ3)

Offspring-A(3Χ4)

Offspring-B(3Χ4)

35

In the first generation, the current population is also the initial population. After

calculating for all the strings in the current population, selection is carried out. The

probability that strings in the current population are copied (i.e. duplicated) and placed in

the intermediate generation is in proportion to their fitness.

Individuals are chosen using "stochastic sampling with replacement" to fill the

intermediate population. A selection process that will more closely match the expected

fitness values is "remainder stochastic sampling." For each string where is greater

than 1.0, the integer portion of this number indicates how many copies of that string are

directly placed in the intermediate population. All strings (including those with less

than 1.0) then place additional copies in the intermediate population with a probability

corresponding to the fractional portion of . For example, a string with = 1.27

places 1 copy in the intermediate population, and then receives a 0.27 chance of placing a

second copy. A string with a fitness of = 0.57 have a 0.57 chance of placing one

string in the intermediate population. Remainder stochastic sampling is most efficiently

implemented using a method known as stochastic universal sampling (It is a technique

used for selecting potentially useful solution for recombination). Assume that the

population is laid out in random order as in a pie graph, where each individual is assigned

space on the pie graph in proportion to fitness. An outer roulette wheel is placed around

the pie with N equally-spaced pointers. A single spin of the roulette wheel will now

simultaneously pick all N members of the intermediate population.

After the selection has been carried out the construction of the intermediate

population is complete and recombination can occur. This can be viewed as creating the

next population from the intermediate population. A crossover is applied to randomly

paired strings with a probability denoted . The population should already be sufficiently

shuffled by the random selection process. Pick a pair of strings. With probability

"recombine" these strings to form two new strings that are inserted into the next

population.

36

Consider the following binary string: 1101001100101101. The string would

represent a possible solution to some parameter optimization problem. New sample

points in the space are generated by recombining two parent strings. Consider this string

1101001100101101 and another binary string, yxyyxyxxyyyxyxxy, in which the values 0

and 1 are denoted by x and y. Using a single randomly-chosen recombination point, 1-

point crossover occurs as follows:

11010 \/ 01100101101

yxyyx /\ yxxyyyxyxxy

Swapping the fragments between the two parents produces the following offspring:

11010yxxyyyxyxxy and yxyyx01100101101. After recombination, we can apply a

mutation operator. For each bit in the population, mutate with some low probability .

Typically the mutation rate is applied with 0.1% - 1% probability. After the process of

selection, recombination and mutation is complete, the next population can be evaluated.

The process of valuation, selection, recombination and mutation forms one generation in

the execution of a genetic algorithm.

5.1.2 Strengths

The power of GAs comes from the fact that the technique is robust and can deal

successfully with a wide range of difficult problems. GAs is not guaranteed to find the

global optimum solution to a problem, but they are generally good at finding "acceptably

good" solutions to problems "acceptably quickly". Where specialized techniques exist for

solving particular problems, they are likely to outperform GAs in both speed and

accuracy of the final result. Even where existing techniques work well, improvements

have been made by hybridizing them with a GA. The basic mechanism of a GA is so

robust that, within fairly wide margins, parameter settings are not critical.

5.1.3 Weaknesses

A problem with GAs is that the genes from a few comparatively highly fit (but not

optimal) individuals may rapidly come to dominate the population, causing it to converge

37

on a local maximum. Once the population has converged, the ability of the GA to

continue to search for better solutions is effectively eliminated: crossover of almost

identical chromosomes produces little that is new. Only mutation remains to explore

entirely new ground, and this simply performs a slow, random search.

5.2 Hybrid Genetic Algorithm

5.2.1 Introduction

The performance of a genetic algorithm, like any global optimization algorithm, depends

on the mechanism for balancing the two conflicting objectives, which are exploiting the

best solutions found so far and at the same time exploring the search space for promising

solutions. The power of genetic algorithms comes from their ability to combine both

exploration and exploitation in an optimal way. However, although this optimal

utilization may be theoretically true for a genetic algorithm, there are problem in practice.

These arise because Holland assumed that the population size is infinite, that the fitness

function accurately reflects the suitability of a solution, and that the interactions between

genes are very small.

In practice, the population size is finite, which influences the sampling ability of a

genetic algorithm and as a result affects its performance. Incorporating a local search

method within a genetic algorithm can help to overcome most of the obstacles that arise

as a result of finite population sizes. Incorporating a local search method can introduce

new genes which can help to combat the genetic drift problem caused by accumulation of

stochastic errors due to finite populations (genetic drift is the random change in the

genetic composition of a population due to chance events causing unequal participation

of individuals in producing succeeding generations. Along with natural selection, genetic

drift is a principal force in evolution). It can also accelerate the search towards the global

optimum, which in turn can guarantee that the convergence rate is large enough to

obstruct any genetic drift.

38

Due to its limited population size, a genetic algorithm may also sample bad

representatives of good search regions and good representatives of bad regions. A local

search method can ensure fair representation of the different search areas by sampling

their local optima, which in turn can reduce the possibility of premature convergence. A

finite population can cause a genetic algorithm to produce solutions of low quality

compared with the quality of solution that can be produced using local search methods.

The difficulty of finding the best solution in the best found region accounted for the

genetic algorithm operator‟s inability to make small moves in the neighborhood of

current solution. Utilizing a local search method within a genetic algorithm can improve

the exploiting ability of the search algorithm without limiting its exploring ability. If the

right balance between global exploration and local exploitation capabilities can be

achieved, the algorithm can easily produce solutions with high accuracy.

Although the genetic algorithms can rapidly locate the region in which the global

optimum exists, they take relatively long time to locate the exact local optimum in the

region of convergence. A combination of genetic algorithm and a local search method

can speed up the search to locate the exact global optimum. In such a hybrid, applying a

local search to the solutions that are guided by a genetic algorithm to the most promising

region can accelerate convergence to the global optimum. The time needed to reach the

global optimum can be further reduced if local search methods and local knowledge are

used to accelerate to locating the most promising search region in addition to locating the

global optimum starting within its basin of attraction.

39

No

Yes

Yes

No

Figure 7: Hybrid genetic algorithm

In the present study the optimal parameters of the dynamic models are determined using

hybrid genetic algorithm. As shown in Figure 7, the algorithm is a combination of the

genetic algorithm and the gradient search (local search method). The hybrid genetic

algorithm steps used for discrete time domain approaches in this study are based on

Upreti and Ein-Mozaffari (2006) presented algorithm. The hybrid genetic algorithm steps

Initial Population

Objective Function

Selection

Crossover

Mutation

Has the generation

limit been exceeded?

Final solution

Optimal vector

Gradient search

Has augmented objective

function changed?

40

used for continuous time domain approach were given by Patel et al. (2007). The

objective function, constraint, mapping, and input variables used in this hybrid genetic

algorithm are given in next section.

Objective function

The objective function used in this simulation is given by:

Mapping

For any optimization parameter, a mapping relates the binary-coded deviation

and the mean parameter value to the parameter value . Thus, a mapping provides

a vector corresponding to each binary-coded deviation vector in its population.

The presented optimization algorithm (Upreti, 2006) uses the following logarithmic and

linear mappings:

Logarithmic Mapping

The purpose of logarithmic mapping is to emphasize relative precision (Coley, 1999b)

within the elements of X. For any optimization parameter, the logarithmic mapping

provides the value, where,

If

If

In Equation 52, b is logarithmic base and and are the maximum and

minimum values of the parameter, . In Equation 53, is the value of the domain

41

between the limits of and , and is the number of representative bits for

any i-th element of i.e. .

Linear Mapping

The linear mapping is straightforward, and given by

Inputs

The presented optimization algorithm needs the following inputs:

(1) The mathematical model and its parameters for the calculation of objective

function;

(2) The number of optimization parameters , and constraints;

(3) The minimum values of control domain, its maximum value

and a factor to vary the size of control domain;

(4) A seed number to generate pseudo-random numbers;

The following parameters for the genetic operations of selection, crossover, and

mutation:

(a) The number of bits for each optimization parameter

(b) The number of cross-over sites for each i.e each optimization parameter

(c) The probability of cross-over

(d) The probability of mutation

(e) The power index to select objective function

(f) The number of genetic generations every iteration

The hybrid optimization algorithm developed in the above section was applied to the

optimization problem for above mentioned three dynamic models. A multi-parameter

mapped coding (Goldberg, 1989) was employed for the binary representation of the

optimization parameters.

42

5.2.2 Steps of algorithm

Following is the hybrid genetic algorithm:

(1) Initialize,

(a) , the vector of mean value of optimization parameters using,

where in equation (24) is the -th pseudo-random number obtained

from a pseudo-random number generator (Knuth, 1973).

(b) a population of binary-coded deviation vectors using the

pseudo- random number generator;

(c) the parameter domain, for each optimization

. parameter.

(d) and based on empirical conditions.

(2) Set logarithmic mapping for the genetic operations of selection, crossover, and

mutation.

(3) Generate an optimal vector by repeating the following consecutive operations on

the population of for generation:

i. Objective function evaluation for each ,

ii. Selection based on the scaled objective function ,

iii. Crossover with probability ,

iv. Mutation with probability ,

(4) Obtain the vector, , and corresponding generated so far using genetic

operators. Set the counter, Set and .

(5) For gradient search, the augmented function based on the interior penalty function

(Rao, 1996) is given by:

for Ein-Mozaffari‟s dynamic model

43

for Patel‟s dynamic model

for Soltanzadeh‟s dynamic model

where is a small positive number.

Equation (56), incorporates the inequality constraints, i.e., equation (16) - (18).



The derivatives required for the gradient search are provided in Appendix. Set

in this step.

(6) Set the gradient search counter, Set and

calculate the

corresponding augmented objective function for the gradient search.

(7) Calculate the vector of the partial derivatives of

, i.e.,

By using equations (20) – (26) for Ein-Mozaffari‟s dynamic model.

By using equations (47) – (50) for Patel‟s dynamic model.

By using equations (37) – (43) for soltanzadeh‟s dynamic model.

If then set and go to step 12.

(8) Calculate along the steepest descent direction as follows:

Where is some positive function. Calculate the corresponding .

44

(9) If

then set , and

and go to step 12.

(10) If

then set and

and go to step 12.

(11) Set and go to step 7.

(12) Calculate

corresponding to If

then set

, and

and go to step 15.

(13) Set

, and If

, or then

go to step 15.

(14) Reduce the penalty term by setting for, Equation (56) of Ein-Mozaffari‟s

dynamic model in discrete time domain, Equation (57) of Patel‟s dynamic model in

continuous time domain, and Equation (58) of Soltanzadeh‟s dynamic model in

discrete time domain. Where is some positive fraction. Set , and go to

Step 6.

(15) Store the resulting optimal value of objective function , and corresponding

optimal vector

(16) Replace by .

(17) Repeat Steps 3 – 16 once with linear mapping.

(18) For each optimization parameter:

i. If is equal to either or , set the size variation factor for control

domain, . (This step allows the alternation of the successive

contraction of with its successive expansion.)

ii. Set . If , set . If set

(This step allows the variation of within its limits.)

(19) Go to Step 2 until the iterative change in is negligible.

45

5.3 Experimental overview

The experimental data used in this work were gathered by Saeed et al. (2008). A

complete system was designed to investigate the continuous-flow mixing process (Figure

8).

Figure 8: Schematic of experimental setup

1 Tracer Tank, 2 Injection pump, 3 Solenoid valve, 4 Computer, 5 Feed tank, 6

Progressive cavity pump, 7 Flow through conductivity sensor and transmitter, 8 Mixing

vessel, 9 Rotary torque transducer with an encoder disk, 10 Motor, 11 Discharge tank.

The major component of the mixing system is the mixing vessel. A flat-bottomed

cylindrical, tank with a volume of 0.075 m3

was used (internal diameter 40 cm and height

60 cm). The tank was equipped with four baffles (4 cm wide and 1.2 cm thick) whose

9

9

3

7

5

8

10

6

2

2

11

2

2

2

4

1

4

6

2

2

46

lengths were equal to the tank height. The tank was supplied with a, 2.5 cm diameter top-

entry shaft driven by a 2 hp motor (Neptune Chemical Pump Co., USA). Impeller torque

was measured with a torque meter (Staiger Mohilo, Germany) equipped with an encoder

disk to measure impeller speed. Impeller speed was also checked with a tachometer.

The second part of the experimental setup was flow system, consisting of feed

section and discharge section. Each section contains a 0.3 m3 cylindrical tank and a

progressive cavity pump (Moyno Industrial Products, USA). Mixing tank liquid level was

kept constant at 41 1 cm. To maintain this constant level, inlet flow-rate was kept

constant and the outlet flow-rate was manipulated to adjust liquid level in the tank.

The third part of the experimental system was the tracer injection unit. To study the

dynamic of continuous mixing, a tracer was injected in the input stream by using a

metering pump (Milton Roy, USA). The tracer was prepared in a 0.06 m3 plastic tank (1).

The injection of the tracer was controlled by a solenoid valve (3) (Ascolectric Ltd.,

Canada). The conductivity variation in inlet and outlet streams of the mixing vessel were

measured using two flow-through conductivity meter.

The last part of the experimental setup was the data acquisition system (4). This

system was able to record impeller torque and both input and output conductivity

readings. Data acquisition system was controlled by using LABVIEW software (National

Instruments, USA).

The xanthan gum solution was pumped from feed tank through a cylindrical mixing

vessel and into the discharge tank. The mixing vessel was equipped with a top entry

impeller. The dynamic tests were performed by injecting a tracer (saline solution)

through a computer controlled solenoid valve into the xanthan gum feed. The

conductivity of input and output streams were measured with flow through conductivity

sensors and analyzed.

47

The process variables used in the experiment were impeller speed, fluid mass

concentration, fluid flow rate, and input/output location. The ranges of experimental

variables used in the simulation are given in Table 1.

Table 1 Variables used in experimental work.

Variables Range

A200 impeller Speed, (rpm) 50 to 700

Xanthan gum mass concentration 0.5%, 1.0%, 1.5%

Xanthan gum flow rate, L/h 227, 603, 896

Input and output location

Configuration 1 Configuration 2

5.4 Rheology of xanthan gum

Xanthan gum is a high-molecular-weight extracellular polysaccharide. It is soluble in

cold water and it has a wide range of applications. Xanthan gum applications are split

approximately 50/50 between food and non food. Non-food applications include oil field,

personal care, pharmaceutical and home care. Typical food applications include sauces

and dressings, baked goods, beverages, desserts and ice creams (Imenson., 2010).

48

The xanthan gum solution exhibit Newtonian viscosity at very low shear rates,

followed by a pseudoplastic region as shear rate increases and, finally, an upper

Newtonian viscosity at very high shear rates. These rheological properties translate into

the very effective suspending and flow properties exhibited by xanthan gum solutions.

Xanthan gum is a very effective thickner and stabilizer compared to other hydrocolloids

(Imenson., 2010).

In this study, parameters of the dynamic models were identified by using the technique

(hybrid genetic algorithm) developed by Upreti and Ein-Mozaffari (2006). The discrete

time domain approach and continuous time domain approach were used for identification.

The simulation was done in three ways: dynamic model with 2 time delays (Ein-

Mozaffari et al., 2004) in discrete time domain, dynamic model with 2 time delays (Patel

et al., 2007) in continuous time domain, and dynamic model with 1 time delay

(Soltanzadeh et al., 2008) in discrete time domain. The constraint used for parameters are

given in Appendix. The sensitivity analysis was done in order to check influence of

parameters on dynamic model (Ein-Mozaffari et al., 2004) results. One-at-a-time analysis

(D. M. Hamby, 1994) method was used to check sensitivity of dynamic model

parameters. The experimental data used in this study were for cylindrical mixing vessel

and pseudoplastic fluids with yield stress material (xanthan gum solutions).

49

CHAPTER 6

RESULTS AND DISCUSSION

Two parameters were used here to quantify the performance of the continuous mixing: f,

the fraction of fluid short-circuiting or channeling in the mixing vessel, and , the

ratio of the fully mixed volume to the total volume of the fluid in the mixing vessel. The

primary mixing occurs in the mixing zone; hence the volume ratio is given by: [Ein-

Mozaffari et al., 2002]

where Q is the fluid flow rate through the mixing vessel and is the total volume of

the fluid in the mixing vessel. The effect of recirculation was not observed in the

dynamic response of laboratory scale mixing, thus R (recirculation) was set to zero in

this study. The hybrid genetic algorithm was employed to compare the dynamic

parameters for the following models using the input-output data:

Dynamic model with 2 time delays in discrete time domain, (Figure 2)

Dynamic model with 2 time delays in continuous time domain, (Figure 3)

Dynamic model with 1 time delay in discrete time domain, (Figure 4)

6.1 Model validation

In this study the dynamic models were validated in two steps. In the first step, we

identified the dynamic model parameters using the first set of data. In the second step,

using the parameters obtained from the first step, we simulated the second dataset in

order to compare the model output with experimental (real system) output. The second

dataset had the same experimental conditions as the first dataset but different input signal.

50

Figure 9 shows the experimental input, experimental output and above mentioned

three dynamic model predicted outputs for the two steps of model validation. The

experimental conditions for the data shown in Figure 9 are: A200 impeller type, 250 rpm

impeller speed, 0.5% mass concentration of xanthan gum, and 227 L/h fluid flow rate.

Figure 9 a and b show that the predicted outputs using the dynamic model with 2 time

delays in discrete time domain (Ein-Mozaffari‟s model) were in good agreement with the

experimental outputs for first and second dataset, respectively. Figure 9 c and d show that

the predicted outputs through the dynamic models with 2 time delays in continuous time

domain (Patel‟s model) were in good agreement with the experimental outputs for first

and second dataset, respectively. Figure 9 e and f show that the predicted outputs using

the dynamic model with 1 time delay in discrete time domain were not in good agreement

with the experimental outputs for first and second dataset, respectively. The predicted

parameters by dynamic models for first set of data are given in Table 2.

Model validation shows that the dynamic models with 2 time delays in both discrete

and continuous time domain were able to represent accurately the dynamic behavior of

the continuous mixing system. While the dynamic model with 1 time delay in discrete

time domain little poorly represented the dynamics of the continuous mixing system. The

extent of channeling predicted by the dynamic model with 1 time delay in discrete time

domain was 15.02% instead of 23.30% (saeed et al., 2008).

Table 2 Predicted parameters by the dynamic models for first set of data.

Parameters

Ein-Mozaffari‟s Model 40 197 0.9893 0.9989 0.2301

Patel‟s Model 40 197 0.9893 0.9989 0.2348

Solatanzadeh‟s Model 60 --- 0.998 0.9988 0.1502

51

(a)

(b)

(c)

0

1

2

3

4

5

0 500 1000 1500 2000 2500

Conduct

ivit

y (

mS

)

Time (s)

5

10

15

20

25

0 500 1000 1500 2000 2500Time (s)

Conduct

ivit

y(m

S)

0

1

2

3

4

5

0 500 1000 1500 2000 2500Time (s)

Conduct

ivit

y(m

S)

52

(d)

(e)

(f)

Figure 9: Predicted outputs using 3-dynamic model in model validation, Experimental

input signal (Dark gray line), Experimental output signal (Gray line), and dynamic model

predicted output (Black line).

0

5

10

15

20

25

0 500 1000 1500 2000 2500Time (s)

Conduct

ivit

y (

mS

)

0

1

2

3

4

5

0 500 1000 1500 2000 2500Time (s)

Con

du

ctiv

ity (

mS

)

0

5

10

15

20

25

0 500 1000 1500 2000 2500Time (s)

Conduct

ivit

y (

mS

)

53

6.2 Model predicted output

Figure 10 shows the experimental input, experimental output, dynamic model predicted

output for above mentioned dynamic models. The experimental conditions for the data

shown in Figure 10 are: A200 impeller type, 150 rpm impeller speed, 1.5% mass

concentration of xanthan gum, and 896 L/h fluid flow rate. Figure 10 shows that the

simulation results of the dynamic models with 2 time delays in both discrete time domain

and continuous time domain were in good agreement with the experimental output, while

the predicted outputs using the dynamic model with 1 time delay were not in good

agreement with the experimental output. The dynamic model with 1 time delay

considered that material takes equal time in both short-circuiting and mixing zones, to

pass from inlet to outlet of the mixing vessel. Because of this assumption, the predicted

output shows deviation from the experimental output. Generally in the short-circuiting

zone fluid leaves the mixing vessel sooner than the fluid in the mixing zone.

The dynamic models with 2 time delays have two separate time delays for short-

circuiting zone and mixing zone, hence they captured well the dynamics of both short-

circuiting and mixing zones. The root mean square error between the models predicted

output and experimental output are: 7.35% for the model with 2 time delays in discrete

time domain, 7.36% for the model with 2 time delays in continuous time domain, and

18.37% for the model with 1 time delay in discrete time domain. When the channeling is

high, the dynamic model with 1 time delay was not predict correctly the extent of short-

circuiting and fully-mixed volume for the continuous mixing systems.

54

(a)

(b)

(c)

Figure 10: Predicted outputs using 3-dynamic models, Experimental input signal (Dark

gray line), Experimental output signal (Gray line), and dynamic model predicted output

(Black line).

1

2

3

4

0 200 400 600 800 1000

Conduct

ivit

y

(mS

)

Time (s)

1

2

3

4

0 200 400 600 800 1000

Conduct

ivit

y

(mS

)

Time (s)

1

2

3

4

0 200 400 600 800 1000

Conduct

ivit

y

(mS

)

Time (s)

55

6.3 Effect of impeller speeds on the extent of channeling and fully mixed volume

Figure 11 shows the results predicted by dynamic model with 2 time delays in discrete

time domain (Ein-Mozaffari‟s model), dynamic model with 2 time delays in continuous

time domain (Patel‟s model), and dynamic model with 1 time delay in discrete time

domain (Soltanzadeh‟s model). The experimental conditions for the data shown in Figure

11 are: A200 impeller type, 1.5% mass concentration of xanthan gum, and 227 L/h fluid

flow rate. It can be seen that f and values computed by the dynamic models with

2 time delays in both discrete time domain and continuous time domain, are in good

agreement with those reported by saeed et al., (2008). However, the results predicted by

dynamic model with 1 time delay deviates from those reported by saeed et al., (2008)

especially at N < 200 rpm.

Figure 11 shows that f (channeling) decreased and the ratio increased as the

impeller speed increased. The mixing zone in the mixing vessel progressively increases

as the impeller speed increases. As a result, the short-circuiting in the mixing vessel

decreases. The results predicted by dynamic models with 2 time delays, in both discrete

and continuous time domain show good agreement with experimental results from higher

to lower impeller speed. However, results predicted by the dynamic model with 1 time

delay show that as the impeller speed increases from100 rpm to 150 rpm and from 200

rpm to 250 rpm, f increases and decreases. We expect that as the impeller speed

increases f decreases and increases. Thus, some of the results obtained from the

dynamic model with 1 time delay contradict those predicted by the dynamic models with

2 time delays. The dynamic model with 1 time delay underestimates f (channeling) and

overestimates especially at lower impeller speed. When the channeling is high,

the dynamic model with 1 time delay was not predict correctly the extent of channeling

and fully-mixed volume for the continuous mixing systems. Table 3 shows the CPU time

taken for simulation, by above mentioned three dynamic models.

56

(a)

(b)

Figure 11: Predicted results using 3-dynamic models at different impeller speeds

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 100 200 300 400 500 600 700

f

Saeed's Data (2008)

Ein-Mozaffari's Model (2004)

Patel's Model (2007)

Soltanzadeh's Model (2008)

Impeller Speed (rpm)

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600 700

Vfm

/ V

t


Saeed's Data (2008)




57

Table 3 Comparison of time taken for simulation by computer (Pentium (R) 4 CPU 3

GHz, 0.99 GB of RAM), For specified search intervals of parameters. The experimental

conditions are: A200 impeller, 1.5% xanthan gum solution, and 227 L/h fluid flow rate.

rpm Model with 1 time

delay in discrete

time domain,

simulation

time in s.

Model with 2 time

delays in discrete

time domain,

simulation

time in s.

Model with 2 time

delays in continuous

time domain,

simulation

time in s.

Sampling

time in s.

50 206.54 386.08 3184.19 2071

100 121.27 344.30 1069.38 2319

150 192.14 398.84 1066.38 2249

200 232.22 483.53 1230.48 2299

250 218.24 400.47 1116.93 2290

300 205.37 416.86 1448.92 2239

400 165.73 383.81 1525.61 2174

500 98.42 373.02 2321.13 2299

600 123.19 313.21 1797.88 2309

700 125.93 292.24 1923.46 2379

58

Table 4 shows the root mean square errors obtained with dynamic model with 1 time

delay in discrete time domain, for given search interval of parameters. The experimental

conditions for the data shown in Table 4 are: A200 impeller type, 1.5% mass

concentration of xanthan gum, and 227 L/h fluid flow rate. Minimum root mean square

error indicates the optimum values of the f (channeling), within given search interval of

parameters (Appendix). Table 4 a shows that for a given experimental conditions, the

dynamic model with 1 time delay in discrete time domain predicted optimum channeling

has the range of minimum root mean square error. Table 4 b shows root mean square

errors obtained with the dynamic model with 1 time delay in discrete time domain, for

different parameters of the genetic algorithm. It can be seen that for given search interval

of parameters, the predicted channeling by the dynamic model with 1 time delay in

discrete time domain are optimum values.

Table 4 Root mean square errors obtained from the dynamic model with 1 time delay in

discrete time domain, for the given search region of parameter f

(a) 0.7, 0.04, 80

RPM f = 0 - 0.2 f = 0.2 -

0.4

f = 0.4 -

0.6

f = 0.6 -

0.8

f = 0.8 -

1.0

Predicted

f

Experimental

f

50 1.270 0.118 1.160 1.170 1.260 0.510 0.715

100 0.367 0.365 3.670 0.367 0.368 0.264 0.642

150 0.465 0.456 4.650 0.465 0.465 0.344 0.594

200 0.315 0.314 3.560 0.359 0.360 0.206 0.492

250 0.285 0.246 2.470 0.294 0.317 0.376 0.409

300 0.564 0.512 5.530 0.820 1.130 0.275 0.336

400 0.156 0.245 5.690 0.946 1.390 0.111 0.188

500 0.613 0.693 1.240 1.770 2.340 0.080 0.117

600 0.577 0.797 1.210 1.700 2.230 0.050 0.028

700 0.487 0.637 1.080 1.560 2.000 0.020 0.003

59

(b) 0.95, 0.04, 120

RPM f = 0 - 1 Predicted

f

Experimental

f

50 1.160 0.516 0.715

100 0.367 0.289 0.642

150 0.465 0.332 0.594

200 0.301 0.205 0.492

250 0.246 0.377 0.409

300 0.512 0.275 0.336

400 0.167 0.117 0.188

500 0.619 0.080 0.117

600 0.577 0.050 0.028

700 0.487 0.020 0.003

6.4 Effect of xanthan gum mass concentration on the extent of channeling and fully

mixed volume

Figure 12 shows the extent of channeling and fully mixed volume predicted by three

dynamic models at different xanthan gum mass concentrations (0.5%, 1.0%, and 1.5%) as

a function of impeller speed. The experimental conditions for the data shown in Figure 12

are: fluid flow rate 603 L/h and different speed of impeller A200. The data reported by

Saeed et al., (2008) for different xanthan gum mass concentrations show that f

(channeling) increased and the ratio decreased as suspension concentration

increased at a fixed impeller speed. Energy delivered by the impeller will be quickly

dissipated in the concentrated xanthan gum solution without producing enough mixing in

the vessel, leading to higher f (channeling) and lower values.

60

(a) 0.5%Xanthan gum solution

(b) 0.5%Xanthan gum solution

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 100 200 300 400 500

f

Saeed's Data (2008)





0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500

Vfm

/ V

t


Saeed's Data (2008)




61

(c) 1.0% Xanthan gum solution

(d) 1.0% Xanthan gum solution

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 100 200 300 400 500 600

fSaeed's Data (2008)





0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600

Vfm

/ V

t


Saeed's Data (2008)




62

(e) 1.5% Xanthan gum solution

(f) 1.5% Xanthan gum solution

Figure 12: Predicted results using 3-dynamic models at different mass concentrations.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 100 200 300 400 500 600 700

f

Saeed's Data (2008)





0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600 700

Vfm

/ V

t


Saeed's Data (2008)




63

Figure 12 b shows that as the impeller speed increased from 150 rpm to 200 rpm,

decreased, based on the results predicted by the dynamic model with 1 time

delay in discrete time domain. Figures 12 c and d shows that as the impeller speed

increased from 50 rpm to 150 rpm, f (channeling) increased and decreased, for

the results obtained from the dynamic model with 1 time delay in discrete time domain.

Figure 12 e shows that as the impeller speed increased from 150 rpm to 200 rpm, f

(channeling) increased based on the results computed by the dynamic model with 1 time

delay in discrete time domain. It can be seen that the dynamic model with 1 time delay

was not predict correctly the extent of f and fully mixed volume as a function of impeller

speed especially at the lower impeller speed. The estimated values by the dynamic model

with 2 time delays in both discrete and continuous time domain show that as the impeller

speed increased the f (channeling) decreased and increased.

Figure 12 shows when mass concentration increases from 0.5% to 1.5%, at a fixed

impeller speed (50 rpm), f (channeling) decreased and increased based on the

results obtained from the dynamic model with 1 time delay in discrete time domain. We

expect that at a specified impeller speed, as the xanthan gum mass concentration

increases, f (channeling) increases and decreases. However the results estimated

by the dynamic model with 1 time delay in discrete time domain contradict those results

reported by saeed et al., (2008). Because of the assumption of the equal time delays, the

dynamic model with 1 time delay underestimates f (channeling) and overestimates

. When the channeling is high, the dynamic model with 1 time delay is not capturing

well the effect of xanthan gum concentration and impeller speed on the dynamics of

continuous mixing. Figure 12 shows that the dynamic models with 2 time delays in both

discrete time domain and continuous time domain were able to predict well the effect of

xanthan gum mass concentration on the dynamic behavior of the continuous flow mixing

system.

64

6.5 Effect of fluid flow rates on the extent of channeling and fully mixed volume

Figures 11, 12 e - f, and 13 shows the extent of channeling and fully mixed volume

predicted by three dynamic models at different flow rates (227 L/h, 603 L/h, and 896 L/h)

as a function of A200 impeller speed at 1.5% xanthan gum concentration. It can be seen

that based on the results reported by saeed et al., (2008), f (channeling) increased and

decreased as the fluid flow rate increased at a fixed impeller speed. Increasing

the flow rate reduces the mean residence time in the vessel, forcing the material to leave

the vessel faster without going into the well-mixed region close to the impeller. Figure

13 a and b show that as the impeller speed increased from 150 rpm to 200 rpm, f

increased and decreased, based on the results computed by the dynamic model

with 1 time delay in discrete time domain. When fluid flow rate increased from 227 L/h

to 603 L/h, Figure 11 and 12 e-f show that at a fixed impeller speed (e.g. 50 rpm and 100

rpm) f decreased and increased, based on the results computed by the dynamic

model with 1 time delay in discrete time domain. We expect that at a fixed impeller

speed, as the fluid flow rate increases, f increases and decreases. However the

results predicted by the dynamic model with 1 time delay in discrete time domain

contradict those reported by saeed et al., (2008).

The extent of f (channeling) and predicted by the models with 2 time delays

in both discrete time domain and continuous time domain show that as the impeller speed

increased f decreased and increased. f and ratio , predicted by the models

with 2 time delays show that as the fluid flow rate increased the f increased and

decreased. When the channeling and fluid concentration are high, the dynamic model

with 1 time delay is not capable of capturing the effect of fluid flow rate and impeller

speed on the dynamics of continuous mixing. Figures 11, 12 e - f, and 13 shows the

predicted results of dynamic models with 2 time delays in both discrete time domain and

continuous time domain, are in good agreements with the data reported by saeed et al.,

(2008).

65

(a)

(b)

Figure 13: Predicted results using 3-dynamic models for 896 L/h flow rate.

.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 100 200 300 400 500 600 700

fSaeed's Data (2008)





0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600 700

Vfm

/ V

t


Saeed's Data (2008)




66

6.6 Effect of different output locations on the extent of channeling and fully mixed

volume

Figure 14 shows the effect of the outlet location on the degree of f and , predicted

by three dynamic models. The outlet locations are: (a) Configuration 1 with bottom outlet

and (b) Configuration 2 with side outlet (Table 1). The experimental conditions for the

data shown in Figure 14 are: xanthan gum mass concentration 0.5%, flow rate 227 L/h,

A200 impeller. Configuration 1 gave the lower value of f and the higher value of

for a given impeller speed. Configuration 2 (where the exit location is on the side of

the vessel) enables a large percentage of feed to be conveyed directly to the exit location,

without being drawn into the mixing zone. In Configuration 1, however, the feed is

forced to flow through the mixing zone before leaving the vessel.

Figures 14 a and b show that as the impeller speed increased from 200 rpm to 250

rpm, f (channeling) increased and decreased based on the results obtained from

the dynamic model with 1 time delay. Figure 14 c and d show that as the impeller speed

increased from 200 rpm to 250 rpm, f (channeling) increased for the results predicted by

the dynamic model with 1 time delay. The dynamic model with 1 time delay

underestimates the extent of f (channeling) and overestimates the extent of . The

predicted values by the dynamic model with 1 time delay show that a large percentage of

feed passed through the mixing zone instead of the short-circuiting zone for

Configuration 2. At impeller speeds 150 rpm and 200 rpm, the predicted results by the

dynamic model with 1 time delay in discrete time domain showed that channeling were

20.57% and 20.01%, respectively, while the results reported by saeed et al., (2008) were

58 % and 44.8%, respectively. Figure 14 shows that the dynamic models with 2 time

delays in discrete time domain and continuous time domain were able to accurately

capture the effect of outlet locations on the dynamic behavior of the continuous mixing

system. While the dynamic model with 1 time delay in discrete time domain was not able

to capture the effect of outlet location.

67

(a) Configuration 1

(b) Configuration 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 100 200 300 400 500

f

Saeed's Data (2008)





0

0.2

0.4

0.6

0.8

1

0 100 200 300 400

Vf

m/

Vt


Saeed's Data (2008)




68

(c) Configuration 2

(d) Configuration 2

Figure 14: Predicted outputs by using 3-dynamic models for different output locations

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 100 200 300 400

f

Saeed's Data (2008)





0

0.2

0.4

0.6

0.8

1

0 100 200 300 400

Vfm

/ V

t


Saeed's Data (2008)




69

6.7 Sensitivity analysis

It was observed that the predicted results by the dynamic models with 2 time delays gave

a better match with the experimental results than those computed by using the dynamic

model with 1 time delay. In order to study the effect of the dynamic model parameters on

the dynamic model response, sensitivity analysis was done for the dynamic model with 2

time delay in discrete time domain. Figure 15 shows the sensitivity analysis plots for the

parameters of the Ein-Mozaffari et al. (2004) dynamic model. The experimental

conditions for the data shown in Figure 15 are: xanthan gum mass concentration 1.0%,

flow rate 603 L/h, A200 impeller, 100 rpm and 500 rpm impeller speed.

Figure 15 a shows that is more sensitive at lower impeller speed (N = 100 rpm)

when mixing is far from ideal. Figure 15 b shows that is more sensitive at higher

impeller speed (N = 500 rpm) when the mixing quality approaches to ideal. It was

observed that as the channeling increases or fully-mixed volume decreases, the sensitivity

of parameter increases, and as the channeling decreases or fully-mixed volume

increases, the sensitivity of parameter increases. Figure 15 c shows that the parameter

is more sensitive in the case of high channeling than low channeling. As the

channeling increases or fully-mixed volume decreases the sensitivity of parameter

increases. Figure 15 d shows that the parameter is more sensitive in the case of low

channeling than high channeling. As the channeling decreases, the sensitivity of

parameter increases on the dynamic model results. It was observed that the parameter

became dominant parameter when mixing approached to an ideal mixing.

.

70

(a)

(b)

0.00E+00

5.00E-02

1.00E-01

1.50E-01

2.00E-01

2.50E-01

3.00E-01

0 10 20 30 40 50 60 70

100 rpm

500 rpm

d1

rms

0.00E+00

5.00E-02

1.00E-01

1.50E-01

2.00E-01

2.50E-01

3.00E-01

0 20 40 60 80 100 120 140 160

100 rpm

500 rpm

rms

d2

71

(c)

(d)

Figure 15: Sensitivity analysis

0.00E+00

5.00E-02

1.00E-01

1.50E-01

2.00E-01

2.50E-01

3.00E-01

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

100 rpm

500 rpm

a1

rms

0.00E+00

5.00E-02

1.00E-01

1.50E-01

2.00E-01

2.50E-01

3.00E-01

0.98 0.985 0.99 0.995 1

100 rpm

500 rpm

a2

rms

73

CHAPTER 7

CONCLUSION AND RECOMMENDATION FOR FUTURE WORK

7.1 Conclusion

The characterization parameters for the continuous mixing of xanthan gum solutions

were determined. The dynamic models with 2 time delays were able to predict well the

effect of process variables (e.g. impeller speed, mass concentration, flow rate, output

location) on the dynamic behavior of the continuous flow mixing system. The dynamic

responses computed using the dynamic models with 2 time delays in both discrete time

domain and continuous time domain showed excellent agreement with the experimental

results.

The simplified dynamic model with 1 time delay in discrete time domain, underestimated

(channeling) and overestimated , especially when the mixing was far from ideal

mixing.

The sensitivity analysis with respect to dynamic model parameters indicated that the

parameters representing the short-circuiting zone were sensitive toward the model

predicted results when the channeling was high. The parameters representing the mixing

zone were sensitive toward the model-predicted results when the continuous mixing of

xanthan gum solutions approached ideal mixing.

74

7.2 Recommendations for future work

The following are suggestions for future investigations:

1. Estimate dynamic model parameters by using standard PEM (prediction-error

minimization) function in MATLAB identification toolbox and compare with the

parameters predicted in this work

2. Determine parameters sensitivity on the dynamic model predicted results by using

other methods for sensitivity analysis

3. Estimate dynamic model parameter by using hybrid genetic algorithm for industrial

data of continuous mixing of non-Newtonian fluids.

75

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79

APPEDIX

Parameter constraints

= 10 – 500, = 10 – 1000, and = 0 - 1

A200 Impeller, 227 l/h flow rate, 1.5% mass concentrations.

rpm

50 0.995 - 0.001 0.99 - 0.0016

100 0.9978 - 0.001 0.99 - 0.0016

150 0.9958 - 0.001 0.99 - 0.0016

200 0.9958 - 0.001 0.99 - 0.0013

250 0.9958 - 0.001 0.99 - 0.0012

300 0.965 - 0.03 0.9988 - 0.0005

400 0.965 - 0.03 0.9988 - 0.0005

500 0.79 - 0.2 0.9988 - 0.0011

600 0.86 - 0.1 0.9987 - 0.0012

700 0.5 - 0.4 0.9987 - 0.0012


rpm

50 0.85 - 0.14 0.995 - 0.0045

100 0.85 - 0.11 0.995 - 0.0045

150 0.93 - 0.05 0.996 - 0.003

200 0.92 - 0.05 0.996 - 0.003

250 0.15 - 0.8 0.9967 - 0.0032

300 0.15 - 0.8 0.9967 - 0.0032

400 0.1 - 0.8 0.9968 - 0.0029

500 0.05 - 0.8 0.9968 - 0.0029

80

A200 Impeller, 603 l/h flow rate, 1.0% mass concentration.

rpm

50 0.98 - 0.01 0.995 - 0.004

100 0.96 - 0.03 0.995 - 0.004

150 0.94 - 0.05 0.995 - 0.004

200 0.94 - 0.05 0.996 - 0.003

250 0.91 - 0.06 0.996 - 0.003

300 0.85 - 0.1 0.9967 - 0.003

400 0.5 - 0.45 0.9968 - 0.003

500 0.55 - 0.3 0.9968 - 0.003

600 0.1 - 0.85 0.9967 - 0.003


rpm

50 0.98 - 0.001 0.9952 - 0.0047

100 0.98 - 0.001 0.99 - 0.003

150 0.95 - 0.03 0.9952 - 0.0047

200 0.95 - 0.024 0.9957 - 0.0042

250 0.91 - 0.05 0.9957 - 0.0032

300 0.95 - 0.04 0.9957 - 0.0032

400 0.95 - 0.04 0.9967 - 0.0031

500 0.94 - 0.05 0.9967 - 0.0031

600 0.8 - 0.1 0.9967 - 0.0032

700 0.8 - 0.1 0.9967 - 0.0032

81


rpm

50 0.99 - 0.005 0.991 - 0.005

100 0.96 - 0.03 0.991 - 0.008

150 0.96 - 0.03 0.991 - 0.008

200 0.9 - 0.09 0.991 - 0.005

250 0.9 - 0.08 0.9953 - 0.003

300 0.9 - 0.08 0.9953 - 0.004

400 0.1 - 0.8 0.9953 - 0.004

500 0.1 - 0.8 0.9953 - 0.004

600 0.1 - 0.8 0.995 - 0.004

700 0.1 - 0.8 0.994 - 0.005

A200 Impeller, 227 l/h flow rate, 0.5% mass concentration for Configuration 1.

rpm

150 0.99 - 0.0065 0.9985 - 0.001

200 0.99 - 0.004 0.998 - 0.001

250 0.99 - 0.008 0.9987 - 0.001

400 0.96 - 0.032 0.9987 - 0.001

A200 Impeller, 227 l/h flow rate, 0.5% mass concentration for Configuration 2.

rpm

150 0.998 - 0.0017 0.99 - 0.0013

200 0.998 - 0.0017 0.99 - 0.0013

250 0.998 - 0.0017 0.99 - 0.0012

400 0.96 - 0.03 0.99 - 0.0012

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